Page:Scientific Papers of Josiah Willard Gibbs.djvu/275

increment of energy of the surface, the above condition expresses that the increment of the total energy of the system is positive. That we have only considered the possible formation of such films as are capable of existing in equilibrium between the given homogeneous masses can not invalidate the conclusion in regard to the stability of the film, for in considering whether any state of the system will have less energy than the given state, we need only consider the state of least energy, which is necessarily one of equilibrium.

If the expression (516) is capable of a negative value for an infinitesimal change in the nature of the part of the film to which the symbols relate, the film is obviously unstable.

If the expression is capable of a negative value, but only for finite and not for infinitesimal changes in the nature of this part of the film, the film is practically unstable^[1] i.e., if such a change were made in a small part of the film, the disturbance would tend to increase. But it might be necessary that the initial disturbance should also have a finite magnitude in respect to the extent of surface in which it occurs; for we cannot suppose that the thermodynamic relations of an infinitesimal part of a surface of discontinuity are independent of the adjacent parts. On the other hand, the changes which we have been considering are such that every part of the film remains in equilibrium with the homogeneous masses on each side; and if the energy of the system can be diminished by a finite change satisfying this condition, it may perhaps be capable of diminution by an infinitesimal change which does not satisfy the same condition. We must therefore leave it undetermined whether the film, which in this case is practically unstable, is or is not unstable in the strict mathematical sense of the term.

Let us consider more particularly the condition of practical stability, in which we need not distinguish between finite and infinitesimal changes. To determine whether the expression (516) is capable of a negative value, we need only consider the least value of which it is capable. Let us write it in the fuller form

$\epsilon ^{S''}-\epsilon ^{S'}-t(\eta ^{S''}-\eta ^{S'})$	$-\mu _{a}(m_{a}^{S''}-m_{a}^{S'})-\mu _{b}(m_{b}^{S''}-m_{b}^{S'})-{\text{etc.}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ (519)
	$-\mu _{g}(m_{g}^{S''}-m_{a}^{S'})-\mu _{h}(m_{h}^{S''}-m_{h}^{S'})-{\text{etc.,}}$

where the single and double accents distinguish the quantities which relate to the first and second states of the film, the letters without accents denoting those quantities which have the same value in both states. The differential of this expression when the quantities distinguished by double accents are alone considered variable, and the area of the surface is constant, will reduce by (501) to the form

$(\mu _{g}''-\mu _{g}')dm_{g}^{S''}+(\mu _{h}''-\mu _{h}')dm_{h}^{S''}+{\text{etc.}}$

1. With respect to the sense in which this term is used, compare page 79.